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Inside Collection (Textbook):

Textbook by: Denny Burzynski, Wade Ellis. E-mail the authors

# Grouping Symbols and the Order of Operations

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses grouping symbols and the order of operations. By the end of the module students should be able to understand the use of grouping symbols, understand and be able to use the order of operations and use the calculator to determine the value of a numerical expression.

## Section Overview

• Grouping Symbols
• Multiple Grouping Symbols
• The Order of Operations
• Calculators

## Grouping Symbols

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:

### ( ), [ ], { }, 

Parentheses: ( )
Brackets: [ ]
Braces: { }
Bar:



In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.

### Sample Set A

If possible, determine the value of each of the following.

#### Example 1

9 + ( 3 8 ) 9 + ( 3 8 ) size 12{9+ $$3 cdot 8$$ } {}

Since 3 and 8 are within parentheses, they are to be combined first.

9+(38) =9+24 =33 9+(38) =9+24 =33

Thus,

9 + ( 3 8 ) = 33 9 + ( 3 8 ) = 33 size 12{9+ $$3 cdot 8$$ ="33"} {}

#### Example 2

( 10 ÷ 0 ) 6 ( 10 ÷ 0 ) 6 size 12{ $$"10"÷0$$ cdot 6} {}

Since 10÷010÷0 size 12{"10" div 0} {} is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."

### Practice Set A

If possible, determine the value of each of the following.

#### Exercise 1

16(32)16(32) size 12{"16"- $$3 cdot 2$$ } {}

10

#### Exercise 2

5+(79)5+(79) size 12{5+ $$7 cdot 9$$ } {}

68

#### Exercise 3

(4+8)2(4+8)2 size 12{ $$4+8$$ cdot 2} {}

24

#### Exercise 4

28÷(1811)28÷(1811) size 12{"28"÷ $$"18"-"11"$$ } {}

4

#### Exercise 5

(33÷3)11(33÷3)11 size 12{ $$"33"÷3$$ -"11"} {}

0

#### Exercise 6

4+(0÷0)4+(0÷0) size 12{4+ $$0÷0$$ } {}

##### Solution

not possible (indeterminant)

## Multiple Grouping Symbols

When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.

### Sample Set B

Determine the value of each of the following.

#### Example 3

2 + ( 8 3 ) ( 5 + 6 ) 2 + ( 8 3 ) ( 5 + 6 ) size 12{2+ $$8 cdot 3$$ - $$5+6$$ } {}

Combine 8 and 3 first, then combine 5 and 6.

2+24-11 Now combine left to right. 26-11 15 2+24-11 Now combine left to right. 26-11 15

#### Example 4

10 + [ 30 ( 2 9 ) ] 10 + [ 30 ( 2 9 ) ] size 12{"10"+ $"30"- $$2 cdot 9$$$ } {}

Combine 2 and 9 since they occur in the innermost set of parentheses.

10+[30-18] Now combine 30 and 18. 10+12 22 10+[30-18] Now combine 30 and 18. 10+12 22

### Practice Set B

Determine the value of each of the following.

#### Exercise 7

(17+8)+(9+20)(17+8)+(9+20) size 12{ $$"17"+8$$ + $$9+"20"$$ } {}

54

#### Exercise 8

(556)(132)(556)(132) size 12{ $$"55"-6$$ - $$"13" cdot 2$$ } {}

23

#### Exercise 9

23+(12÷4)(112)23+(12÷4)(112) size 12{"23"+ $$"12"÷4$$ - $$"11" cdot 2$$ } {}

4

#### Exercise 10

86+[14÷(108)]86+[14÷(108)] size 12{"86"+ $"14"÷ $$"10"-8$$$ } {}

93

#### Exercise 11

31+{9+[1+(352)]}31+{9+[1+(352)]} size 12{"31"+ lbrace 9+ $1+ $$"35"-2$$$ rbrace } {}

74

#### Exercise 12

{6[24÷(42)]}3{6[24÷(42)]}3 size 12{ lbrace 6- $"24"÷ $$4 cdot 2$$$ rbrace rSup { size 8{3} } } {}

27

## The Order of Operations

Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of 3+523+52 size 12{3+5 cdot 2} {}. We could do either of two things:

Add 3 and 5, then multiply this sum by 2.

3+52 =82 =16 3+52 =82 =16

Multiply 5 and 2, then add 3 to this product.

3+52 =3+10 =13 3+52 =3+10 =13

We now have two values for one number. To determine the correct value, we must use the accepted order of operations.

### Order of Operations

1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order 2, 3, 4 described below,
2. Perform all exponential and root operations.
3. Perform all multiplications and divisions, moving left to right.
4. Perform all additions and subtractions, moving left to right.

### Sample Set C

Determine the value of each of the following.

#### Example 5

21+312 Multiply first. 21+36 Add. 57 21+312 Multiply first. 21+36 Add. 57

#### Example 6

(15-8)+5(6+4). Simplify inside parentheses first. 7+510 Multiply. 7+50 Add. 57 (15-8)+5(6+4). Simplify inside parentheses first. 7+510 Multiply. 7+50 Add. 57

#### Example 7

63-(4+63)+76-4 Simplify first within the parenthesis by multiplying, then adding. 63-(4+18)+76-4 63-22+76-4 Now perform the additions and subtractions, moving left to right. 41+76-4 Add 41 and 76:41+76=117. 117-4 Subtract 4 from 117: 117-4=113. 113 63-(4+63)+76-4 Simplify first within the parenthesis by multiplying, then adding. 63-(4+18)+76-4 63-22+76-4 Now perform the additions and subtractions, moving left to right. 41+76-4 Add 41 and 76:41+76=117. 117-4 Subtract 4 from 117: 117-4=113. 113

#### Example 8

7642+15 Evaluate the exponential forms, moving left to right. 7616+1 Multiply 7 and 6:76=42 4216+1 Subtract 16 from 42:4216=26 26+1 Add 26 and 1:26+1=27 27 7642+15 Evaluate the exponential forms, moving left to right. 7616+1 Multiply 7 and 6:76=42 4216+1 Subtract 16 from 42:4216=26 26+1 Add 26 and 1:26+1=27 27

#### Example 9

6(32+22)+42 Evaluate the exponential forms in the parentheses: 32=9 and 22=4 6(9+4)+42 Add the 9 and 4 in the parentheses:9+4=13 6(13)+42 Evaluate the exponential form:42=16 6(13)+16 Multiply 6 and 13:613=78 78+16 Add 78 and 16:78+16=94 94 6(32+22)+42 Evaluate the exponential forms in the parentheses: 32=9 and 22=4 6(9+4)+42 Add the 9 and 4 in the parentheses:9+4=13 6(13)+42 Evaluate the exponential form:42=16 6(13)+16 Multiply 6 and 13:613=78 78+16 Add 78 and 16:78+16=94 94

#### Example 10

62+2242+622+13+82102195 Recall that the bar is a grouping symbol. The fraction62+2242+622is equivalent to(62+22)÷(42+622) 36 + 4 16 + 6 4 + 1 + 64 100 19 5 36 + 4 16 + 24 + 1 + 64 100 95 40 40 + 65 5 1 + 13 14 62+2242+622+13+82102195 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } + { {1 rSup { size 8{3} } +8 rSup { size 8{2} } } over {"10" rSup { size 8{2} } -"19" cdot 5} } } {} Recall that the bar is a grouping symbol. The fraction62+2242+622 size 12{ { {6 rSup { size 8{2} } +2 rSup { size 8{2} } } over {4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} } } } } {}is equivalent to(62+22)÷(42+622) size 12{ $$6 rSup { size 8{2} } +2 rSup { size 8{2} }$$ ÷ $$4 rSup { size 8{2} } +6 cdot 2 rSup { size 8{2} }$$ } {} 36 + 4 16 + 6 4 + 1 + 64 100 19 5 size 12{ { {"36"+4} over {"16"+6 cdot 4} } + { {1+"64"} over {"100"-"19" cdot 5} } } {} 36 + 4 16 + 24 + 1 + 64 100 95 size 12{ { {"36"+4} over {"16"+"24"} } + { {1+"64"} over {"100"-"95"} } } {} 40 40 + 65 5 size 12{ { {"40"} over {"40"} } + { {"65"} over {5} } } {} 1 + 13 size 12{1+"13"} {} 14 size 12{"14"} {}

### Practice Set C

Determine the value of each of the following.

#### Exercise 13

8+(327)8+(327) size 12{8+ $$"32"-7$$ } {}

33

#### Exercise 14

(34+1823)+11(34+1823)+11 size 12{ $$"34"+"18"-2 cdot 3$$ +"11"} {}

57

#### Exercise 15

8(10)+4(2+3)(20+315+405)8(10)+4(2+3)(20+315+405) size 12{8 $$"10"$$ +4 $$2+3$$ - $$"20"+3 cdot "15"+"40"-5$$ } {}

0

#### Exercise 16

58+422258+4222 size 12{5 cdot 8+4 rSup { size 8{2} } -2 rSup { size 8{2} } } {}

52

#### Exercise 17

4(6233)÷(424)4(6233)÷(424) size 12{4 $$6 rSup { size 8{2} } -3 rSup { size 8{3} }$$ ÷ $$4 rSup { size 8{2} } -4$$ } {}

3

#### Exercise 18

(8+93)÷7+5(8÷4+7+35)(8+93)÷7+5(8÷4+7+35) size 12{ $$8+9 cdot 3$$ ÷7+5 cdot $$8÷4+7+3 cdot 5$$ } {}

125

#### Exercise 19

33+236229+582+247232÷83+1823333+236229+582+247232÷83+18233 size 12{ { {3 rSup { size 8{3} } +2 rSup { size 8{3} } } over {6 rSup { size 8{2} } -"29"} } +5 left ( { {8 rSup { size 8{2} } +2 rSup { size 8{4} } } over {7 rSup { size 8{2} } -3 rSup { size 8{2} } } } right )÷ { {8 cdot 3+1 rSup { size 8{8} } } over {2 rSup { size 8{3} } -3} } } {}

7

## Calculators

Using a calculator is helpful for simplifying computations that involve large num­bers.

### Sample Set D

Use a calculator to determine each value.

#### Example 11

9, 842 + 56 85 9, 842 + 56 85 size 12{9,"842"+"56" cdot "85"} {}

 Key Display Reads Perform the multiplication first. Type 56 56 Press × 56 Type 85 85 Now perform the addition. Press + 4760 Type 9842 9842 Press = 14602

#### Example 12

42 ( 27 + 18 ) + 105 ( 810 ÷ 18 ) 42 ( 27 + 18 ) + 105 ( 810 ÷ 18 ) size 12{"42" $$"27"+"18"$$ +"105" $$"810"÷"18"$$ } {}

 Key Display Reads Operate inside the parentheses Type 27 27 Press + 27 Type 18 18 Press = 45 Multiply by 42. Press × 45 Type 42 42 Press = 1890

Place this result into memory by pressing the memory key.

 Key Display Reads Now operate in the other parentheses. Type 810 810 Press ÷ 810 Type 18 18 Press = 45 Now multiply by 105. Press × 45 Type 105 105 Press = 4725 We are now ready to add these two quantities together. Press + 4725 Press the memory recall key. 1890 Press = 6615

Thus, 42(27+18)+105(810÷18)=6,61542(27+18)+105(810÷18)=6,615 size 12{"42" $$"27"+"18"$$ +"105" $$"810"÷"18"$$ =6,"615"} {}

#### Example 13

16 4 + 37 3 16 4 + 37 3 size 12{"16" rSup { size 8{4} } +"37" rSup { size 8{3} } } {}

 Nonscientific Calculators Key Display Reads Type 16 16 Press × 16 Type 16 16 Press × 256 Type 16 16 Press × 4096 Type 16 16 Press = 65536 Press the memory key Type 37 37 Press × 37 Type 37 37 Press × 1396 Type 37 37 Press × 50653 Press + 50653 Press memory recall key 65536 Press = 116189
 Calculators with yxyx Key Key Display Reads Type 16 16 Press y x y x size 12{y rSup { size 8{x} } } {} 16 Type 4 4 Press = 4096 Press + 4096 Type 37 37 Press y x y x size 12{y rSup { size 8{x} } } {} 37 Type 3 3 Press = 116189

Thus, 164+373=116,189164+373=116,189 size 12{"16" rSup { size 8{4} } +"37" rSup { size 8{3} } ="116","189"} {}

We can certainly see that the more powerful calculator simplifies computations.

#### Example 14

Nonscientific calculators are unable to handle calculations involving very large numbers.

85612 21065 85612 21065 size 12{"85612" cdot "21065"} {}

 Key Display Reads Type 85612 85612 Press × 85612 Type 21065 21065 Press =

This number is too big for the display of some calculators and we'll probably get some kind of error message. On some scientific calculators such large numbers are coped with by placing them in a form called "scientific notation." Others can do the multiplication directly. (1803416780)

### Practice Set D

Use a calculator to find each value.

#### Exercise 20

9,285+86(49)9,285+86(49) size 12{9,"285"+"86" $$"49"$$ } {}

13,499

#### Exercise 21

55(8426)+120(512488)55(8426)+120(512488) size 12{"55" $$"84"-"26"$$ +"120" $$"512"-"488"$$ } {}

6,070

#### Exercise 22

10631741063174 size 12{"106" rSup { size 8{3} } -"17" rSup { size 8{4} } } {}

1,107,495

#### Exercise 23

6,05336,0533 size 12{6,"053" rSup { size 8{3} } } {}

##### Solution

This number is too big for a nonscientific calculator. A scientific calculator will probably give you 2.217747109×10112.217747109×1011 size 12{2 "." "217747109"´"10" rSup { size 8{"11"} } } {}

## Exercises

For the following problems, find each value. Check each result with a calculator.

### Exercise 24

2+3(8)2+3(8) size 12{2+3 cdot $$8$$ } {}

26

### Exercise 25

18+7(41)18+7(41) size 12{"18"+7 cdot $$4-1$$ } {}

### Exercise 26

3+8(62)+113+8(62)+11 size 12{3+8 cdot $$6-2$$ +"11"} {}

46

### Exercise 27

15(88)15(88) size 12{1-5 cdot $$8-8$$ } {}

### Exercise 28

3716237162 size 12{"37"-1 cdot 6 rSup { size 8{2} } } {}

1

### Exercise 29

98÷2÷7298÷2÷72 size 12{"98"÷2÷7 rSup { size 8{2} } } {}

### Exercise 30

(4224)23(4224)23 size 12{ $$4 rSup { size 8{2} } -2 cdot 4$$ -2 rSup { size 8{3} } } {}

0

### Exercise 31

9+149+14 size 12{ sqrt {9} +"14"} {}

### Exercise 32

100+8142100+8142 size 12{ sqrt {"100"} + sqrt {"81"} -4 rSup { size 8{2} } } {}

3

### Exercise 33

83+82583+825 size 12{ nroot { size 8{3} } {8} +8-2 cdot 5} {}

### Exercise 34

1641+521641+52 size 12{ nroot { size 8{4} } {"16"} -1+5 rSup { size 8{2} } } {}

26

### Exercise 35

6122+4[3(10)+11]6122+4[3(10)+11] size 12{"61"-"22"+4 $3 cdot $$"10"$$ +"11"$ } {}

### Exercise 36

1214[(4)(5)12]+1621214[(4)(5)12]+162 size 12{"121"-4 cdot $$$4$$ cdot $$5$$ -"12"$ + { {"16"} over {2} } } {}

97

### Exercise 37

(1+16)37+5(12)(1+16)37+5(12) size 12{ { { $$1+"16"$$ -3} over {7} } +5 cdot $$"12"$$ } {}

### Exercise 38

8(6+20)8+3(6+16)228(6+20)8+3(6+16)22 size 12{ { {8 cdot $$6+"20"$$ } over {8} } + { {3 cdot $$6+"16"$$ } over {"22"} } } {}

29

### Exercise 39

10[8+2(6+7)]10[8+2(6+7)] size 12{"10" cdot $8+2 cdot $$6+7$$$ } {}

### Exercise 40

21÷7÷321÷7÷3 size 12{"21"÷7÷3} {}

1

### Exercise 41

1023÷523231023÷52323 size 12{"10" rSup { size 8{2} } cdot 3÷5 rSup { size 8{2} } cdot 3-2 cdot 3} {}

### Exercise 42

85÷558585÷5585 size 12{"85"÷5 cdot 5-"85"} {}

0

### Exercise 43

5117+7251235117+725123 size 12{ { {"51"} over {"17"} } +7-2 cdot 5 cdot left ( { {"12"} over {3} } right )} {}

### Exercise 44

223+23(62)(3+17)+11(6)223+23(62)(3+17)+11(6) size 12{2 rSup { size 8{2} } cdot 3+2 rSup { size 8{3} } cdot $$6-2$$ - $$3+"17"$$ +"11" $$6$$ } {}

90

### Exercise 45

2626+20132626+2013 size 12{"26"-2 cdot left lbrace { {6+"20"} over {"13"} } right rbrace } {}

### Exercise 46

2{(7+7)+6[4(8+2)]}2{(7+7)+6[4(8+2)]} size 12{2 cdot lbrace $$7+7$$ +6 cdot $4 cdot $$8+2$$$ rbrace } {}

508

### Exercise 47

0+10(0)+15{43+1}0+10(0)+15{43+1} size 12{0+"10" $$0$$ +"15" cdot lbrace 4 cdot 3+1 rbrace } {}

### Exercise 48

18+7+2918+7+29 size 12{"18"+ { {7+2} over {9} } } {}

19

### Exercise 49

(4+7)(83)(4+7)(83) size 12{ $$4+7$$ cdot $$8 - 3$$ } {}

### Exercise 50

(6+8)(5+24)(6+8)(5+24) size 12{ $$6+8$$ cdot $$5+2 - 4$$ } {}

144

### Exercise 51

(213)(61)(7)+4(6+3)(213)(61)(7)+4(6+3) size 12{ $$"21" - 3$$ cdot $$6 - 1$$ cdot $$7$$ +4 $$6+3$$ } {}

### Exercise 52

(10+5)(10+5)4(604)(10+5)(10+5)4(604) size 12{ $$"10"+5$$ cdot $$"10"+5$$ - 4 cdot $$"60" - 4$$ } {}

1

### Exercise 53

628+3(5)(2)+84+(1+8)(1+11)628+3(5)(2)+84+(1+8)(1+11) size 12{6 cdot left lbrace 2 cdot 8+3 right rbrace - $$5$$ cdot $$2$$ + { {8} over {4} } + $$1+8$$ cdot $$1+"11"$$ } {}

### Exercise 54

25+3(8+1)25+3(8+1) size 12{2 rSup { size 8{5} } +3 cdot $$8+1$$ } {}

52

### Exercise 55

34+24(1+5)34+24(1+5) size 12{3 rSup { size 8{4} } +2 rSup { size 8{4} } cdot $$1+5$$ } {}

### Exercise 56

16+08+52(2+8)316+08+52(2+8)3 size 12{1 rSup { size 8{6} } +0 rSup { size 8{8} } +5 rSup { size 8{2} } cdot $$2+8$$ rSup { size 8{3} } } {}

25,001

### Exercise 57

(7)(16)34+22(17+32)(7)(16)34+22(17+32) size 12{ $$7$$ cdot $$"16"$$ - 3 rSup { size 8{4} } +2 rSup { size 8{2} } cdot $$1 rSup { size 8{7} } +3 rSup { size 8{2} }$$ } {}

### Exercise 58

2375223752 size 12{ { {2 rSup { size 8{3} } - 7} over {5 rSup { size 8{2} } } } } {}

#### Solution

125125 size 12{ { {1} over {"25"} } } {}

### Exercise 59

1+62+236+11+62+236+1 size 12{ { { left (1+6 right ) rSup { size 8{2} } +2} over {3 cdot 6+1} } } {}

### Exercise 60

621233+43+2325621233+43+2325 size 12{ { {6 rSup { size 8{2} } - 1} over {2 rSup { size 8{3} } - 3} } + { {4 rSup { size 8{3} } +2 cdot 3} over {2 cdot 5} } } {}

14

### Exercise 61

58296257+724224558296257+7242245 size 12{ { {5 left (8 rSup { size 8{2} } - 9 cdot 6 right )} over {2 rSup { size 8{5} } - 7} } + { {7 rSup { size 8{2} } - 4 rSup { size 8{2} } } over {2 rSup { size 8{4} } - 5} } } {}

### Exercise 62

(2+1)3+23+11062152252552(2+1)3+23+11062152252552 size 12{ { { $$2+1$$ rSup { size 8{3} } +2 rSup { size 8{3} } +1 rSup { size 8{"10"} } } over {6 rSup { size 8{2} } } } - { {"15" rSup { size 8{2} } - left [2 cdot 5 right ] rSup { size 8{2} } } over {2 rSup { size 8{4} } - 5} } } {}

0

### Exercise 63

63210222+18(23+72)2(19)3363210222+18(23+72)2(19)33 size 12{ { {6 rSup { size 8{3} } - 2 cdot "10" rSup { size 8{2} } } over {2 rSup { size 8{2} } } } + { {"18" $$2 rSup { size 8{3} } +7 rSup { size 8{2} }$$ } over {2 $$"19"$$ - 3 rSup { size 8{3} } } } } {}

### Exercise 64

26+10262526+102625 size 12{2 cdot left lbrace 6+ left ["10" rSup { size 8{2} } - 6 sqrt {"25"} right ] right rbrace } {}

152

### Exercise 65

1813236+36431813236+3643 size 12{"181" - 3 cdot left (2 sqrt {"36"} +3 nroot { size 8{3} } {"64"} right )} {}

### Exercise 66

28112534210+2228112534210+22 size 12{ { {2 cdot left ( sqrt {"81"} - nroot { size 8{3} } {"125"} right )} over {4 rSup { size 8{2} } - "10"+2 rSup { size 8{2} } } } } {}

#### Solution

4545 size 12{ { {4} over {5} } } {}

### Exercises for Review

#### Exercise 67

((Reference)) The fact that 0 + any whole number = that particular whole number is an example of which property of addition?

#### Exercise 68

((Reference)) Find the product. 4,271×6304,271×630 size 12{4,"271" times "630"} {}.

2,690,730

#### Exercise 69

((Reference)) In the statement 27÷3=927÷3=9 size 12{"27" div 3=9} {}, what name is given to the result 9?

#### Exercise 70

((Reference)) What number is the multiplicative identity?

1

#### Exercise 71

((Reference)) Find the value of 2424 size 12{2 rSup { size 8{4} } } {}.

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##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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